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Properties

Label 2-180-4.3-c8-0-41
Degree $2$
Conductor $180$
Sign $0.648 + 0.760i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.5 − 6.70i)2-s + (166. + 194. i)4-s − 279.·5-s − 2.62e3i·7-s + (−1.10e3 − 3.94e3i)8-s + (4.06e3 + 1.87e3i)10-s + 2.30e3i·11-s + 4.70e4·13-s + (−1.76e4 + 3.81e4i)14-s + (−1.03e4 + 6.47e4i)16-s + 5.19e4·17-s + 5.95e4i·19-s + (−4.64e4 − 5.44e4i)20-s + (1.54e4 − 3.34e4i)22-s + 7.75e4i·23-s + ⋯
L(s)  = 1  + (−0.907 − 0.419i)2-s + (0.648 + 0.760i)4-s − 0.447·5-s − 1.09i·7-s + (−0.270 − 0.962i)8-s + (0.406 + 0.187i)10-s + 0.157i·11-s + 1.64·13-s + (−0.458 + 0.993i)14-s + (−0.158 + 0.987i)16-s + 0.622·17-s + 0.456i·19-s + (−0.290 − 0.340i)20-s + (0.0658 − 0.142i)22-s + 0.277i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.648 + 0.760i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ 0.648 + 0.760i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.291761976\)
\(L(\frac12)\) \(\approx\) \(1.291761976\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (14.5 + 6.70i)T \)
3 \( 1 \)
5 \( 1 + 279.T \)
good7 \( 1 + 2.62e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.30e3iT - 2.14e8T^{2} \)
13 \( 1 - 4.70e4T + 8.15e8T^{2} \)
17 \( 1 - 5.19e4T + 6.97e9T^{2} \)
19 \( 1 - 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 - 7.75e4iT - 7.83e10T^{2} \)
29 \( 1 + 9.02e5T + 5.00e11T^{2} \)
31 \( 1 + 3.40e5iT - 8.52e11T^{2} \)
37 \( 1 + 5.84e5T + 3.51e12T^{2} \)
41 \( 1 + 2.93e5T + 7.98e12T^{2} \)
43 \( 1 - 2.95e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.03e6iT - 2.38e13T^{2} \)
53 \( 1 - 7.54e6T + 6.22e13T^{2} \)
59 \( 1 - 8.82e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.08e7T + 1.91e14T^{2} \)
67 \( 1 + 1.44e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.71e6iT - 6.45e14T^{2} \)
73 \( 1 + 3.62e7T + 8.06e14T^{2} \)
79 \( 1 + 4.88e7iT - 1.51e15T^{2} \)
83 \( 1 - 6.93e7iT - 2.25e15T^{2} \)
89 \( 1 - 1.05e8T + 3.93e15T^{2} \)
97 \( 1 - 1.33e8T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92754691958629844089708583336, −10.16250318240614796969952235935, −9.040995332675689681292609724326, −7.987872862224089613563026246575, −7.27639591120626528246966737415, −6.05172575963088516683970950124, −4.07707616471614442107473545605, −3.34996259261804965640602455041, −1.57134397144234180453777790213, −0.65761207285554488508294210113, 0.74234132971433029000553316725, 2.04938764151784687246258936184, 3.48935005914338037166745789940, 5.33247126066773250463405560357, 6.16273205411821728029180048924, 7.33823892919286194228348754753, 8.556904270156255616938360144975, 8.900172155027740271731114658804, 10.23707349564627983299557026718, 11.23991704566700621513775258193

Graph of the $Z$-function along the critical line